Theorem 1

Tag 00Y2, permanent. Cite it as [Bourbaki, Tag 00Y2].

Lie Groups and Lie Algebras, chapter VIII, § 1. The Lie algebra sl(2, k) and its representations

Every simple $\mathfrak{s}\mathfrak{l}(2, k)$ -module of finite dimension $n$ is isomorphic to $V(n-1)$ . Every finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$ -module is a direct sum of submodules isomorphic to the modules $V(m)$ .

This follows from Lemma 4 and Prop. 1, 2 and 3.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Remark, § 1
Proposition 6, § 1
Theorem 2, § 1
Lemma 2, § 6

Cites

Lemma 4
Prop. 1